For the optimization problem to be efficient, the objective should be convex with respect to the decision variables [1]}. As optimizing with respect to parameters of the dynamics (e.g. impedance parameters) is not standard, we check the convexity of a toy cost function with respect to impedance parameters, denoted \(\phi = [\mathbf {M}, \mathbf {B}]\) . As impedance parameters \(\phi \)  mostly affect covariance, we consider a simplified objective in a single step shooting problem of \(\min _\phi \mathrm {Tr}(\mathbf {Q}\mathbf {\Sigma }^+)\) , where by results from [2]},
\(\mathrm {Tr}(\mathbf {Q}\mathbf {\Sigma }^+) & = & \mathrm {Tr}(\mathbf {Q}\mathbf {A}_\phi \mathbf {\Sigma } \mathbf {A}_\phi ^T + \mathbf {Q}\mathbf {B}_\phi \mathbf {\Sigma }^f \mathbf {B}_\phi ^T)  \\& = & \mathbf {\sigma }^T\mathrm {vec}(\mathbf {A^TQA})+(\mathbf {\sigma }^f)^T\mathrm {vec}(\mathbf {B^TQB})\nonumber \\& = & \left(\mathbf {\sigma }^T(\mathbf {A}^T\mathtt {\otimes }\mathbf {A}^T)\mathtt {+}(\mathbf {\sigma }^f)^T(\mathbf {B}^T\mathtt {\otimes }\mathbf {B}^T\right)\mathrm {vec}(\mathbf {Q}).\nonumber \) 
